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ρ[n_, L_]:=
  FixedPoint[
    ArrayFlatten,TensorProduct@@Table[Subscript[ρ, l,m][k],{k,n},{l,L},{m,L}]];

list2[n_, L_]:=
  ParallelTable[ρ[n,L][[i,j]]->a[i,j],{i,L^n},{j,L^n}]/.
   {a[i_,j_]:>ToExpression["a"<>ToString[i]]}//Flatten;

p1[n_, L_] := 
  Sum[
    Subscript[ρ, L, L][k]*
    Product[
     If[k == k1, 1, 
         ParallelSum[Subscript[ρ, l, l][k1], {l, L - 1}]], {k1, n}], {k, n}]//Expand;

P1[n_, L_]:= P1[n,L]= p1[n,L] /. list2[n,L];

p0[n_, L_]:=
  ParallelProduct[
    Sum[Subscript[ρ, l, l][k], {l, L - 1}], {k, n}] // Expand;

P0[n_, L_]:=P0[n,L]= p0[n,L] /. list2[n,L];

Easy to evaluate...

P0[4, 3]-P0[3, 3]
P1[4, 3]-P1[3, 3]

What kind of changes are needed to speed up this calculation...

P0[5, 3]-P0[4, 3]
P1[5, 3]-P1[4, 3]
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1 Answer 1

By using "Rho[n_, L_]:=Rho[n_,L_]=..." command. We can make it faster...

Rho[n_, L_]:=Rho[n_,L_]=FixedPoint[ArrayFlatten,TensorProduct@@Table[Subscript[\[Rho], l,m][k],{k,n},{l,L},{m,L}]];

list2[n_, L_]:=list2[n_, L_]=ParallelTable[Rho[n,L][[i,j]]->a[i,j],{i,L^n},{j,L^n}]/.{a[i_,j_]:>ToExpression["a"<>ToString[i]]}//Flatten;
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